| L(s) = 1 | − 16·13-s + 20·25-s + 72·37-s − 4·49-s − 64·61-s − 16·97-s − 72·109-s − 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4.43·13-s + 4·25-s + 11.8·37-s − 4/7·49-s − 8.19·61-s − 1.62·97-s − 6.89·109-s − 7.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.995777697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.995777697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 2 p T^{2} + 51 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 26 T^{2} + 531 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 16 T^{2} + 1362 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 80 T^{2} + 3138 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 9 T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 10 T^{2} + 2787 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 62 T^{2} + 4443 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 70 T^{2} + 2187 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 44 T^{2} + 2646 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 70 T^{2} + 5787 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 16 T + 180 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 44 T^{2} + 3318 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 110 T^{2} + 4923 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 262 T^{2} + 30075 T^{4} + 262 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 116 T^{2} + 5382 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.96356768244291460123816524908, −2.95279987940488461058959726773, −2.93986100037202406913883720451, −2.86338300770114560825383493215, −2.85217292964628235219593144575, −2.80200742545663678989121902775, −2.64110285411410659860404076209, −2.57860132009858808037705670405, −2.41956412658744092732424882135, −2.41809082446386980084621520154, −2.34035872810408318314180758878, −2.23799074721963651375769684095, −2.05211161776278149583071426150, −1.59713539203354763884373306138, −1.56488273980748129107658878565, −1.51267564013525909546234150124, −1.33541270929710765694681835280, −1.19241163964774588089922152663, −1.16258443914382381030516145491, −1.10649070392080508803323222944, −0.793123381242227450985761166101, −0.69583990142221778454823065342, −0.38409991216973001025554956948, −0.27461725918572087729231263113, −0.16263994893559228941018338420,
0.16263994893559228941018338420, 0.27461725918572087729231263113, 0.38409991216973001025554956948, 0.69583990142221778454823065342, 0.793123381242227450985761166101, 1.10649070392080508803323222944, 1.16258443914382381030516145491, 1.19241163964774588089922152663, 1.33541270929710765694681835280, 1.51267564013525909546234150124, 1.56488273980748129107658878565, 1.59713539203354763884373306138, 2.05211161776278149583071426150, 2.23799074721963651375769684095, 2.34035872810408318314180758878, 2.41809082446386980084621520154, 2.41956412658744092732424882135, 2.57860132009858808037705670405, 2.64110285411410659860404076209, 2.80200742545663678989121902775, 2.85217292964628235219593144575, 2.86338300770114560825383493215, 2.93986100037202406913883720451, 2.95279987940488461058959726773, 2.96356768244291460123816524908
Plot not available for L-functions of degree greater than 10.
